Statistical Methods in Medical Research 4ed

The above considerations are in terms of controlling the error rate per study.
If a set of variables represents an overall latent characteristic, such as a battery of
tests designed to assess neuropsychological function, then this may be a reasonable
thing to do. In other cases different characteristics may be examined within
a single study, which may be regarded as a set of studies which have been
combined for convenience and economy. Then it is not desirable to control the
error rate of the whole study, since this would reduce the power for each
characteristic according to other characteristics.
Hotelling's T 2 tests
Hotelling's two-sample T 2 statistic
Usually we are interested in possible differences in the mean values of the
variates, and the appropriate test is based on Hotelling's two-sample T 2 statistic.
This assumes that the data follow a multivariate normal distribution, but is
reasonably robust against departures from normality.
Using the same notation as before, calculate
T 2 ˆ nA nB
nA ‡ nB
D 2 : …13:12†
This is the square of the highest possible value of t that could be obtained in a
two-sample t test for any linear combination of the p variables. If the null
hypothesis is true,
…nA ‡ nB p 1† 2
T
…nA ‡ nB 2†p
…13:13†
follows the F distribution on p and nA ‡ nB p 1 degrees of freedom (DF). If
p ˆ 1, (13.13) reduces to the square of the usual t statistic.
In Example 13.2, there is no real point in carrying out a significance test,
since Fig. 13.2 shows clearly that the two bivariate distributions differ significantly.
However, as an illustration of the calculations, we have
T 2 ˆ
63 16
79
13.3 Discriminant analysis 473
4 709 ˆ 60 08
and, since p ˆ 2, 76T 2 =…77 2† ˆ29 65 is referred to the F distribution on
2 and 76 DF, giving P < 0 001. The difference is highly significant, as
expected. In most applications of Hotelling's T 2 test, it is useful to explore
further any significant result by calculating the discriminant function to show
the best way in which the variates combine to reveal the difference between the
groups.